Optimal barbell force-velocity profiles can contribute to maximize weightlifting performance

Maximal barbell power output (Pmax) and vertical barbell threshold velocity (vthres) are major determinants of weightlifting performance. Moreover, an optimal force-velocity relationship (FvR) profile is an additional variable that has the potential to maximize sports performance. The aims of this study were (i) to present a biomechanical model to calculate an optimal FvR profile for weightlifting, and (ii) to determine how vthres, Pmax, and the optimal FvR profile influence theoretical snatch performance (snatchth). To address these aims, simulations were applied to quantify the respective influence on snatchth. The main findings confirmed that at constant vthres and Pmax, snatchth is maximized at an optimal FvR profile. With increasing Pmax and decreasing vthres, the optimal FvR profile becomes more force dominated and more effective to enhance snatchth. However, sensitivity analysis showed that vthres and Pmax have a larger effect on snatchth than the optimal FvR profile. It can be concluded that in weightlifting, training protocols should be designed with the goal to improve Pmax and to reduce vthres to ultimately enhance snatchth. Training programs designed to achieve the optimal FvR profile may constitute an additional training goal to further develop weightlifting performance in elite athletes that already present high Pmax levels.


Introduction
In ballistic sports such as track and field or weightlifting, performance is ultimately determined by the athlete´s capability to maximally accelerate their body mass (e.g., sprinting, jumping) or a maximal external load (i.e., weightlifting) [1,2]. In all of these examples, the velocity reached at the end of the propulsive phase is related to mechanical output parameters produced through efficient work of the neuromuscular system [3]. In this context, a linear force-velocity relationship (FvR) has frequently been used to assess mechanical parameters such as the theoretical maximal velocity at zero force (v 0 ) and the theoretical maximal force at zero velocity (F 0 ). From v 0 and F 0 , maximal power output (P max ) and the FvR profile (i.e., slope of the FvR; s FvR ) can be computed [4]. Previous research has shown that P max is a major determinant of maximal ballistic performance [5,6]. In addition, Samozino  that for a given P max level, performance in vertical jumping and sprinting is (theoretically) maximized at an optimal FvR profile [4,7,8]. Accordingly, a customized resistance training program to improve ballistic sports performance should aim to maximize P max through optimization of the FvR profile. Although the benefits of an optimal FvR profile for practical use are still under debate [9], there is partial evidence that resistance training programs designed to optimize the FvR profile were more successful than standard (i.e., non-optimized) resistance training protocols in improving vertical jump height in trained soccer, rugby, and futsal player [10][11][12].
In weightlifting, maximal performance is determined by the lifters neuromuscular capabilities to produce a high power output combined with well-developed lifting technique (e.g., turnover and catch phase) [13]. The technical mastery and effectiveness of the lift is reflected by the individual's vertical threshold velocity (v thres ) [14]. The v thres is the minimum peak vertical barbell velocity (v max ) an individual athlete needs to lift a maximal barbell load successfully in the overhead position. For example, the v max during a one-repetition maximum (1RM) lift in the snatch is denoted as v thres in the snatch. In general, v thres ensures a necessary vertical travel distance and flight time of the maximal barbell load (i.e., projectile motion) at the end of the acceleration phase, which allows the athlete to squat under and catch the barbell in the overhead position.
Consequently, weightlifters who have better technical skills in the turnover and catch phase can lift with lower levels of v thres . For example, better technical skill performance is indicated through larger amounts of "residual work" and a shorter time-span of the turnover phase [14,15]. term "residual work" has previously been defined as the vertical distance the barbell travels beyond the theoretical distance from projectile motion (i.e., v max ) due to an acting vertical force component that is initiated through the upper extremities [14]. In addition, a low v thres corresponds to a smaller amount of force that needs to be utilized to accelerate the barbell. Hence, higher force levels can be achieved to overcome gravitational forces which results in better weightlifting performance. As recently presented, weightlifting is a good example to use FvR-parameters (i.e., F 0 , v 0 , P max ) to monitor progression during training and to predict weightlifting performance [16,17]. In agreement with results from vertical jumping [4], for individual time-series data P max has been shown to be highly-but not perfectly (i.e., cross-correlation coefficients range from 0.86-0.88)-associated with the theoretical snatch performance (snatch th ) in elite weightlifters [17]. Due to the imperfect correlation of snatch th and P max , the specific FvR profile was assumed to be another determinant of weightlifting performance [17] as it has been previously presented for the vertical jump [4]. Accordingly, a given P max level in combination with an optimal FvR profile may maximize snatch performance.
Therefore, the main aim of this study was to determine how the optimal FvR profile (if any), v thres , and P max influence weightlifting performance. Considering the aforementioned relation between maximal vertical jump performance and an optimal FvR profile [7], we hypothesized that for a given level of P max and v thres, the FvR profile has an impact on the theoretical snatch performance (H1), and that the theoretical snatch performance is maximized at an optimal FvR profile (H2). Further, we were interested to elucidate the extent to which P max , the FvR profile, and v thres influence snatch th . With reference to the literature [4,14], we hypothesized that changes in P max and v thres influence snatch th to a larger degree compared with changes of the FvR profile (H3).
To address these aims, first, we present the biomechanical concept of optimal FvR profile applied to weightlifting (i.e., snatch pull model) from which the maximal theoretical snatch performance (snatch max th ) can be computed. In the second part, we used mathematical simulations to apply the snatch pull model with data from the literature. The simulations were used to quantify the influence of the optimal FvR profile, v thres , and P max on snatch th .

Theoretical background
This section is dedicated to the theory of optimal FvR profile and how an optimal FvR profile may positively influence weightlifting performance. For this purpose, we briefly recap the established biomechanical model from vertical jump and transfer it to weightlifting.
In a linear modelled FvR profile from loaded vertical jumps, P max is located at 0.5F 0 and 0.5v 0 , respectively [3]. Consequently, P max can be calculated as: In this context, the vertical force at P max (i.e., 0.5F 0 ) is associated with an external load condition during the jump. For a given FvR profile, the load at P max conditions is interpreted as the optimal load [18]. In case of a vertical jump, when P max is located at a load that corresponds to the jumpers body mass (i.e., body mass = optimal load) an optimal FvR profile is achieved [4]. Under optimal FvR profile conditions (i.e., P max is located at body mass), the vertical takeoff velocity of the body´s center of mass-and hence the maximal jump height-is maximal. In contrast, even at constant P max level, with a non-optimal FvR profile (i.e., P max is located at loads smaller or larger then body mass), the achieved vertical take-off velocity is less than the maximal vertical take-off velocity at an optimal FvR profile. In other words, with a non-optimal FvR profile, the actual external mechanical power-output during a jump is less than the jumper's maximal power capacities (P max ) [19]. The concept of delivering P max at the load condition corresponding to the targeted movement (here unloaded vertical jump) can be used as a starting point to find an optimal FvR profile that maximize performance in weightlifting.

Transferring theory from jumping to establishing optimal barbell FvR profiles in weightlifting
Performance in weightlifting is simply defined as the maximal load that can be lifted in the snatch and the clean and jerk [20]. However, a successful maximal lift requires acceleration of the barbell load up to an individual v thres during the acceleration phase [2,14]. As previously outlined for the vertical jump [21], performance in weightlifting can also be described by two interacting constraints: i) movement specific barbell velocity conditions (i.e., v thres in m�s -1 ) (black line in Fig 1A) and ii) mechanical output produced by the neuromuscular system (i.e., barbell FvR; dashed black line in Fig 1A).
In this context, it has recently been shown, that for the individual weightlifter, the theoretical snatch performance (snatch th in kg) can be calculated from the linear snatch pull FvR profile and the known v thres of a 1RM snatch [16]. For example, v thres is a highly individual constant that can be obtained from 1RM snatches in competitions using video-based analysis of barbell kinematics. The interaction of the two aforementioned mechanical constraints can be visualized by the intersection of the two lines, giving the mean barbell force at v thres ( � F thres in N) (black dot in Fig 1A) from which snatch th can be calculated. According to the approach outlined by Sandau et al. [16], to compute snatch th , first � F thres needs to be computed from v thres , v 0 (in m�s -1 ), and s FvR (in m�s -1 �N -1 ) as follows: Of note, as adjustment to the approach proposed by Sandau et al. [16], in Eq (2) v 0 and s FvR were extracted from a snatch pull FvR modelled with maximal (instead of mean) vertical barbell velocity (ordinate) and mean vertical barbell force (abscissa). Therefore, the slope of the FvR is calculated as: � F thres represents the sum of the barbell force due to gravity (g in m�s -2 ) and the barbell force due to the mean vertical barbell acceleration to achieve v thres (� a thres in m�s -2 ). Consequently, snatch th is obtained by: In Eq (3), � a thres can be calculated using v thres and the vertical distance of barbell acceleration (i.e., h acc in m; vertical height of the barbell at the instance of v thres minus the radius of barbell plates [0.225 m]): Substitution Eqs (1) and (4) in Eq (3), and after simplification, snatch th can be expressed as: To express snatch th as a function of s FvR and P max , using Eqs (1) and (3), v 0 can be calculated as: v 0 ¼ 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi À P max s FvR p : ð7Þ Finally, substituting Eq (7) in Eq (6) As mentioned above, maximal ballistic performance is achieve, when the actual external power-output during a movement equals P max . For the individual weightlifter, the actual vertical power output during snatch th is highly associated with the absolute load of snatch th [17]. Since power is the product of force and velocity, using Eq (2), the actual vertical power output during snatch th (P snatch th in W) can be calculated as: With � F thres being the mean vertical barbell force at snatch th (Eq (2)). Substituting Eq (2) in Eq (9) gives: As can be seen for the exemplary barbell FvR profile in Fig 1A, P snatch th is not located at P max . Following the same mechanical principle presented for the vertical jump, a mismatch of P snatch th and P max can be interpreted as a non-optimal barbell FvR profile that results in a nonmaximal snatch th . In this case, only a fraction of P max can be used to accelerate the barbell load to v thres . In turn, an optimal barbell FvR profile that enables the weightlifter to accelerate the barbell with the maximal possible vertical barbell power output (P max ) may result in a maximized snatch th performance (snatch max th ). Again, even with a given P max level, snatch th is maximized at an optimal s FvR (s opt FvR ). Graphically, this s opt FvR is achieved, if P max is exactly located at P snatch th (Fig 1B). In the presented example, s opt FvR is reached when P max is shifted towards P snatch th (to the right) by an increase in � F 0 and a decrease in v 0 . In other words, the non-optimal s FvR in this case is caused by a force "deficit" or a velocity "surplus", respectively. For example, if P max needs to be shifted to the left to match P snatch th , the non-optimal s FvR is caused by a velocity "deficit" or a force "surplus", respectively.
As previously mentioned for the optimal FvR profile in vertical jumps, under s opt FvR conditions, v thres equals 0:5v opt o and � F opt thres equals 0:5 � F opt 0 . Since v thres is an individual known constant for a 1RM snatch lift, v opt o can be simply calculated as: Using Eqs (2) and (3), and after simplification, the s opt FvR can be calculated as: Finally, snatch max th is obtained by substituting Eq (12) in Eq (8).

Model simulation
In the theoretical background, we presented a biomechanical model from which the optimal FvR profile (i.e., s opt FvR ) and the maximal theoretical snatch th (snatch max th ) can be computed. This model is based on a snatch pull linear two-point FvR (i.e., only two loading conditions were applied during the snatch pull test) that is obtained using linear regression in eight elite male weightlifters [22]. In order to illustrate how changes in P max and v thres may moderate s opt FvR and snatch max th , typical ranges of P max and v thres were applied for simulations using Eq (8). Furthermore, the simulations were used to quantify the respective contribution of P max , s FvR , and v thres on snatch th .
For the applied simulations, no new experimental data were collected. Instead, we used data from previous (published) studies of our research group [16,17,22]. In total, these experiments were conducted with 10 elite male and 3 elite female weightlifters, modeling the barbell FvR profile and snatch th using the aforementioned snatch pull test. From these experiments, typical value for P max range from 2000 to 4000 W, for s FvR from -0.0005 to -0.0025 m�s -1 �N -1 , and h acc was on average 0.8 m [16,17,22]. In addition, the magnitudes of maximal vertical barbell velocities during the 1RM snatch (i.e., v thres ) can also be obtained from the literature. Typical values for v thres during the snatch 1RM range from 1.70 to 2.0 m�s -1 [15,23,24].
First, changes in snatch max th and s opt FvR were analyzed for different P max (at constant value of v thres ) and v thres values (constant value of P max ), and as a variation of both variables. Second, the influence of v thres , P max , and s FvR on snatch th were analyzed through sensitivity analysis. Within the sensitivity analysis, the relative (i.e., percentage) variation of each independent variable was plotted against the relative snatch th change to assess the relative importance of each variable. Although h acc has an influence on snatch th , this variable was treated as a constant in the simulations as it depends on the athlete's anthropometric characteristics that cannot be influenced through training.

Influence of P max and v thres on snatch th and s FvR
The simulated influence of P max and v thres on snatch th and s FvR is depicted in Fig 2. Findings from the simulation study showed that both P max and v thres influence snatch th . Furthermore, at high P max or a low v thres levels, changes in s FvR have a larger potential to moderate snatch th due to the more prominent apex of the snatch th -s FvR -function (Fig 2). As mathematically presented, snatch th is maximized (i.e., snatch max th ) at an optimal value of s FvR (i.e., s opt FvR , red lines in Fig 2). In fact, it is obvious that with increasing P max and decreasing v thres , s opt FvR is shifted towards a more force dominated FvR profile (Figs 2 and 3).
Since v opt 0 equals 2v thres (Eq (11)), under optimal FvR profile conditions, v opt 0 is a constant and does not depend on the absolute value of snatch th . Therefore, in theory, improvements in weightlifting performance solely may depend on increased theoretical maximal vertical barbell force capabilities (i.e., � F opt 0 ) (Fig 4). This relation results in an improved force at v thres and thus a higher barbell load that can be lifted. According to the example in Fig 4, at a constant v thres of 2.0 m�s -1 , an increase in � F opt 0 by +500 N is associated with an increase in P max of +500 W that corresponds to an increase in snatch max th of about +20 kg.

Relative contribution of P max , v thres , and s FvR on snatch th
The sensitivity analysis showed that relative snatch th performance is primarily influenced by P max and v thres rather than s FvR (Fig 5). Of note, Fig 5 shows that P max and v thres have a continuous positive or negative effect on snatch th , while s FvR has its maximal effect at s opt FvR .

Discussion
This study aimed to introduce the theoretical base of an optimal barbell snatch pull FvR profile (i.e., s opt FvR ) to maximize theoretical snatch 1RM performance (snatch max th ) in weightlifting. We additionally examined the influence of s opt FvR , P max , and v thres on snatch th performance. In line with our study hypotheses, we observed at constant levels of P max or v thres that snatch th is influenced by s FvR and maximized at an optimal s FvR value (i.e., s opt FvR ). We further confirmed that changes in P max and v thres have a larger influence on snatch th than changes in s FvR . Competitive weightlifting requires both well developed technical skills and high mechanical power output to lift a maximal load in the overhead position in the snatch and the clean and jerk [25,26]. The snatch pull FvR is an approach to quantify the external mechanical output at the barbell and to assess training related changes in v 0 , � F 0 , P max and v thres . While the parameters v 0 , � F 0 , P max are related to neuromuscular capabilities, v thres is related to sport-specific technical skills. Our study findings revealed that besides training-induced improvements in neuromuscular capabilities, well-developed technical skills (i.e., amount of v thres ) are needed and affect snatch performance. From the perspective of the neuromuscular capabilities, P max presented the largest contribution to increase weightlifting performance. The leading effect of P max to enhance ballistic performance is in agreement with evidence from the literature [27]. In addition, with an increased performance level (i.e., high P max ), an optimal barbell FvR profile becomes more relevant to further maximize weightlifting performance. In this context, our simulations showed that the optimal barbell FvR profile is primarily force driven as P max increases. This finding seems reasonable, given that the snatch 1RM is highly associated with the weightlifter´s maximal strength capabilities (i.e., 1RM squat) [25,26,28].
From the perspective of weightlifting technical skills, the simulations showed that changes in v thres have a large influence on snatch th . For instance, if two lifters have the same P max level to accelerate a maximal barbell load, the lifter with lower v thres will achieve the higher snatch performance. In this context, Richter [29] postulated that increased maximal muscle strength and power have a larger impact to improve weightlifting performance than increased technical skills (ratio 10:1). In theory, however, if improved technical skills are associated with lowered v thres , the aforementioned ratio is � 1:1.5 ( Fig 5). Nevertheless, the large potential influence of v thres on snatch performance should be put in perspective, since technical skills in weightlifting level off after 4-5 years of systematic training [30]. Therefore, elite weightlifters with a long history of systematic training (>5 years) are more likely to benefit from increased muscle strength and power (i.e., � F 0 , P max ) to improve snatch performance than from lowering v thres . A few limitations of this study should be acknowledged. First, we should mention that the present findings are based on a biomechanical model that was verified using simulations. Although the concept of an optimal FvR to maximize performance seems to work for the vertical jump [10,11], the validity for practical application in weightlifting has not yet been shown. Second, although the measurement error of FvR parameters (i.e., � F 0 , v 0 , P max ) derived from the snatch pull test is rather small [22], the measurement error for s opt FvR and snatch max th has not been assessed yet. However, knowledge of measurement error for s opt FvR and snatch max th is essential to guide training programming for the individual athlete. Finally, the modelled changes in snatch performance are based solely on changes in mechanical parameters during the acceleration phase (lift-off until maximal vertical barbell velocity) without accounting for the effect of an increased barbell load on the execution of the subsequent movement phases (i.e., turnover, catch, stand up) that also limit the performance outcome.

Conclusions
Weightlifting performance can be improved through an increase in mechanical power output (P max ), improved technical skills (v thres ) and optimized FvR profile (s opt FvR ). During long-term athlete development as well as in elite sports, improvements of P max should be the main focus of training to further develop weightlifting performance [13,17,31]. In addition, a well-developed snatch technique enables weightlifters to efficiently lift loads at low v thres . This point can be declared as a major goal during the early stages of the long-term athlete development process [30]. For weightlifters with high P max levels, the contribution of s opt FvR to maximize snatch performance becomes more relevant. This finding is of great importance for elite weightlifters as the contribution of increased P max and lowered v thres to improve performance is strongly limited in world class athletes. Therefore, designing elite weightlifters training to achieve an optimal FvR profile has the potential to maximize performance. Even if optimization of the FvR profile can provide only small improvements in weightlifting performance, it may be of importance in competitions. As pointed out by Sandau and Lenz [32], on average, only 1.0% (� 3.7 kg) of total weightlifting performance (sum of snatch and clean and jerk) was the difference of making it to the podium or not (3 rd vs. 4 th place) at the Olympic Games.